Coding method and apparatus using quaternary codes

ABSTRACT

In digital data transmission and storage, binary data strings are encoded into symbols from a quaternary alphabet by a specific inventive use of finite-state machines. The novel, high-rate quaternary codes offer spectral shaping properties together with a significant increase in noise margin when used on channels which lend themselves to partial-response shaping. In principle, the encoding occurs in three steps: a u-state transition diagram generates quaternary symbols whose running digital sum assumes values from a given set; the u-state transition diagram is converted into a v-state machine, each of the v states being associated with a number of transitions sufficient to encode the input data bytes into output code words; finally, the v-state machine is switched into a next state depending on its current state and the last encoded input data byte.

BACKGROUND OF THE INVENTION

The present invention relates generally to the encoding of data strings, here the encoding of sequential binary data, for digital transmission or recording systems. In particular, the novel coding technique employs high-rate quaternary codes that offer advantageous properties for transmission over metallic cables or optical fibers and for recording in storage media. Further, the present invention is directed to an encoder utilizing the novel coding technique and to new codes, and to a transmission or recording system employing the above method of encoding and the new codes.

In the prior art, many digital transmission or recording systems require the use of codes which impose restrictions on the channel input sequences. Such codes, often termed constrained, modulation, or channel codes, are mainly used to improve timing and gain control in the receiver (or decoder in a recording system), to reduce intersymbol interference, and to shape the spectrum of the transmitted or recorded sequences so that the spectrum matches the frequency characteristics of the channel. In particular, many line codes were designed to suppress the spectral components of encoded sequences near the zero frequency. Such codes are instrumental in reducing the effects of baseline wander in a receiver.

Codes of that kind are DC-free, i.e., have a first order spectral null at zero frequency, if, and only if, the running digital sum (RDS) is bounded. The number of values N that the RDS can assume determines the low frequency content of the code spectrum. For example, the modified version of the ternary MS43 line code described by P. A. Franaszek in "Sequence-State Coding for Digital Transmission", Bell System Technical Journal, Vol. 47, pp. 143-157, January 1968, which was adopted by the German Bundespost as a standard for an ISDN transceiver, is based on a bounded RDS constraint with N=6.

Binary and multilevel codes with a spectral null at zero frequency possess desirable distance properties in addition to their spectrum shaping properties and can therefore be used to improve the reliability of transmission over noisy channels. In particular, they can be used to increase the Euclidian distance at the output of partial-response channels which are often encountered in digital recording and wire transmissions.

Binary Codes for partial-response channels that expand bandwidth by using convolutional codes with good Hamming Distance properties have been considered, e.g., by J. K. Wolf and G. Ungerboeck in "Trellis Codes for Partial Response Channels", IEEE Trans. Commun., Vol. COM-34, pp. 765-773, August 1986. Binary codes that expand bandwidth by enforcing a spectral null have been given, e.g., in U.S. Pat. No. 4,888,775 by R. Karabed and P. H. Siegel, issued in 1989. All these coding schemes are binary, and are well suited for applications such as saturation recording.

Multilevel coding schemes, as discussed, e.g., by G. D. Forney and A. R. Calderbank in "Coset Codes for Partial Response Channels; or Coset Codes with Spectral Nulls", IEEE Trans. Inform. Theory, Vol. IT-35, pp. 925-943, September 1989, achieve higher immunity against noise by expanding the signal alphabet and using trellis or known coset codes adapted for partial-response channels. Low rate (R less than or equal to 1 bit/symbol) quaternary trellis codes for partial response channels have also been constructed using concatenated coding schemes and set partitioning of the channel output signal set. Multilevel coding schemes are well suited for applications such as bandwidth efficient digital transmission and AC-bias recording.

OBJECTS OF THE INVENTION

A primary object of the present invention is to provide an optimal and advantageous method and apparatus for storing or transmitting data using optimized quaternary codes.

A further object of the present invention is to provide a method and means for constructing and applying optimized quaternary codes in the above apparatus. These optimized codes offer the spectral shaping properties of line codes, in particular for baseband wire/cable transmission.

Another object of the present invention is to increase/improve the noise margin during transmission or storage, especially in partial response class IV channels.

A still further object of the present invention is to reduce the average transmitted power at the channel input.

SUMMARY OF THE INVENTION

A method of encoding and decoding data according to the principles of the present invention meeting these and other objects comprises forming a v-state machine according to a v-state transition diagram and generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram wherein the running digital sum of the sequences of the second set assumes values taken from a finite set of integers, such that each of v states is associated with 2K transitions, each transition corresponding to an m-symbol code word; encoding in said v-state machine a k-bit data byte into an m-symbol quaternary code word and generating a next state of said v-state machine as a function of its current state and the last encoded k-bit data byte. Sequences of m-symbol quaternary code words are decoded by estimating the encoded k-bit data bytes from output samples received from a partial response channel by generating the most likely sequence of m-symbol words that can be derived from the u-state transistion diagram, and mapping said generated m-symbol codes words into k-bit data bytes. The present invention also describes encoder/decoder apparatus for performing the disclosed encode/decode methods.

The invention meets these and other objects by the methods stated in claims 1 and 2 and the apparatus defined in claims 6 and 7. preferred embodiments and details of the methods and apparatus according to the invention are defined in the respective subclaims.

In brief, the quaternary coding technique disclosed here differs from conventional multilevel coding schemes for partial-response channels in that it optimizes the data transfer by applying a novel encoding method, i.e., a novel selection process for the code in which the data are to be transmitted or stored. The resulting transmitted (or stored) code exhibits spectral nulls at all frequencies where the channel transfer function vanishes.

Advantages are, first, less errors since the Euclidian distance at the channel output is increased; second, lower power consumption; and third, a relatively simple decoder design, including, e.g., a Viterbi detector.

BRIEF DESCRIPTION OF THE DRAWING

In the following, the preferred embodiment of the invention shall be described in detail with reference to the appended drawing, starting with the mathematical background, discussion of simulation results, and implementations, and in which:

FIG. 1 is an infinite-state transition diagram illustrating quaternary sequences having a spectral null at DC;

FIG. 2 illustrates a first u-state transition diagram according to the present invention;

FIG. 3 illustrates a first v-state transition diagram according to the present invention;

FIG. 4 illustrates a second u-state transition diagram according to the present invention;

FIG. 5 illustrates a second v-state transition diagram according to the present invention;

FIG. 6 illustrates a third u-state transition diagram according to the present invention;

FIG. 7 illustrates a third v-state transition diagram according to the present invention;

FIG. 8 is a graph illustrating the event error probability as a function of the energy-per-symbol to noise-power-density ratio; 2E_(s) /RN_(n), scaled by the rate loss;

FIG. 9 is a graph illustrating the power spectral density of various codes generated according to the present invention;

FIG. 10 is a block diagram illustrating a first encoder implementation according to the present invention;

FIG. 11 is a block diagram illustrating a first decoder implementation according to the present invention;

FIG. 12 is a block diagram illustrating a second encoder implementation according to the present invention; and

FIG. 13 is a block diagram illustrating a second decoder implementation according to the present invention.

THE TABLES

The tables form part of the description and arc, for the sake of convenience, appended to the following text that describes the invention in detail, and in which:

Table 1 lists the capacity C (bit/symbol) for finite-state subdiagrams of the infinite-state transition diagram shown in FIG. 1;

Table 2 specifies the encoder mapping for the encoder of FIG. 3;

Table 3 lists code words in hexadecimal presentation for the encoder of FIG. 3;

Table 4 describes a 10-state trellis structure for Viterbi-detecting the code of Table 3;

Table 5 specifies the encoder mapping for the encoder of FIG. 5;

Table 6 lists code words in hexadecimal representation for the encoder of FIG. 5;

Table 7 describes an 8-state trellis structure for Viterbi-detecting the code of Table 6;

Table 8 specifies the encoder mapping for the encoder of FIG. 7;

Tables 9A/9B list code words in hexadecimal presentation for the encoder ofFIG. 7;

Table 10 describes a 12-state trellis structure for Viterbi-detecting the code of Table 9; and

Table 11 provides a code comparison.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT Coding Scheme

The codes given in this section are based on Finite-State Transition Diagrams (FSTDs) whose edges are labeled by symbols from the quaternary alphabet {-3,-1,+1,+3}. Referring now to FIG. 1 an infinite-state transition diagram whose subdiagrams generate quaternary sequences with bounded RDS, i.e., with a spectral null at DC is illustrated. Let G_(N) be a finite-state subdiagram of this infinite-state transition diagram obtained by restricting the state set to N consecutive states. In this case, the number of states in G_(N) and the number of distinct values assumed by the RDS are the same. Table 1 contains the capacity C, i.e., the maximum achievable code rate, associated with G_(N) for different values of N.

The communication system model consists of a transmitter, a partial-response channel and a receiver. The transmitter employs finite-state machine encoders which map binary data streams into quaternary sequences. The set of all possible encoded sequences is a subset of the set of all sequences generated by the FSTD on which the encoder is based.

The receiver utilizes a soft-decision Viterbi detector followed by a block decoder which inverts the encoder mapping. The Viterbi detector performs sequence estimation on the trellis that combines the FSTD and the partial-response channel trellis. In other words, it chooses the most likely sequence generated by the FSTD in the sense that when it is transmitted over a noise-free channel, it results in a string of symbols which has a minimum Euclidean distance to the noisy received sequence. Thecomplexity of the resulting Viterbi detectors can be kept within manageablelimits without sacrificing performance by exploiting the periodic nature ofquaternary FSTDs.

The construction of efficient codes is usually based on a technique known as state splitting. In the most common form of state splitting, referred to as output state splitting, the outgoing transitions (or edges) of a split state are distributed among the two resulting offsprings, whereas the incoming transitions of the split state are duplicated. A sequence of such splittings results in an equivalent FSTD where the number of outgoingtransitions from each state is now sufficient to perform rate k/m bit/symbol coding. This increase in outgoing transitions has been achievedat the expense of an increase in decoder look-ahead. In other words, the sliding window of the sliding block decoder has become larger. In the construction of the disclosed quaternary codes, state splitting has been applied in a reverse manner. The incoming transitions of a split state aredistributed among the two resulting offsprings whereas the outgoing transitions of the split state are duplicated. Therefore, rather than increasing the number of outgoing transitions, this technique, referred toas input state splitting, allows redirection of transitions without incurring any increase in decoder look-ahead. As a consequence, quasi-catastrophic error propagation can be avoided without climinating a large number of low-power code words and/or increasing decoder look-ahead.Keeping as many as possible low-power code words has the advantage of reducing the average transmitted power.

In general, the p-th power of G_(N) is an N-state transition diagram where a transition from state i to state j corresponds to a p-step path from state i to state j in G_(N) and vice versa The codes disclosed hereare based on 3-state transition diagrams derived from the second power (p=2) of G₇ and G₁₁. In this case, each transition is labeled with two quaternary symbols.

Codes based on G₇

The maximum rate that encoders derived from G₇ N=7, can achieve, is 1.566 bit/symbol (see Table 1). In this case, the number of identical symbols at the input and output of the dicode channel is restricted to 6 and 5, respectively.

FIG. 2 shows a u-state transition diagram, i.e, the 3-state irreducible component of the second power of G₇ from which one of the 8B/6Q codes(rate R=8/6 bit/symbol) is derived. The third power of the FSTD in FIG. 2 gives rise to a new 3-state transition diagram where each state has at least 256 outgoing 6-symbol transitions allowing the realisation of a 3-state 8B/6Q encoder.

One observes that it is possible to start at different encoder states and generate exactly the same sequences. Such sequences give rise to the phenomenon of quasi-catastrophic error propagation and should be eliminated. This can be achieved by eliminating transitions that start andend at the same state and cause error propagation, and rearranging transitions via input state splitting of the center state. In other words,transitions are rearranged such that edges arriving at the center state arepartitioned and redirected to its two offsprings and the outgoing edges areduplicated. This leads to a 4-state transition diagram that does not generate sequences which could give rise to quasi-catastrophic error propagation. Since there are more than 256 transitions emanating from eachof the 4 states, one can discard the transitions that contribute most to the average transmitted power. The reduced average transmitted power associated with sequences generated by this encoder is thus P_(s) =2.93 which is considerably smaller than P_(s) =5, i.e., the average transmitted power of a reference 2B/1Q scheme.

The structure of a resulting v-state encoder (v=4), often referred to as a v-state machine, is shown in FIG. 3. The encoder mapping is specified in Table 2 and, finally, the list of code words in hexadecimal representationis given in Table 3. The two quaternary symbols corresponding to a hexadecimal number are obtained by grouping the binary representation of ahexadecimal number into 2 dibits and converting every dibit 00, 01, 10, 11 into -3, -1, +1, +3, respectively. As an example, the code word D99 has the binary representation 110110011001 and corresponds to the 6 symbols +3-1+1-1+1-1 at the input of the channel. The bar on a list of i code words such as L(i) indicates that the signs of code word symbols in the original list of code words L(i) have been inverted.

For example, +1-1+1-1+1-1 (D99) is in A(43) (see Table 3) and represents the label of one of the 43 transitions in A(43) from state 1 to state 2 orfrom state 2 to state 4 (see FIG. 3). The inverse of D99 is -3+1-1+1-1+1 and belongs to A(43). Table 2 consists of two subtables specifying the mapping of 8-bit data words into 6-symbol code words. Subtable 2.1 provides data word to code word assignment when the encoder is in state 1 and state 4. Subtable 2.2 does the same for the encoder states 2 and 3. The entries of the first column in both subtables indicate lists of data words d in decimal form. For example, the entry d=0-42 in the first columnof Subtable 2.1 indicates the data words 00000000=0, 00000001=1, . . . , 00101010=42. The entries of the second and third columns in both tables are of the form L(i)/s where L(i) is a list of i code words and s=1,2,3,4 indicates the next encoder state. For example, Δ(43)/2 in the first column of Subtable 2.1 indicates a list of 43 code words that can be generated when the encoder changes its state from state s=1 (present state) to state s=2 (next state).

Decoding is accomplished in two steps. First, a Viterbi detector with a path memory of 24 symbols operating every two symbol intervals on the 10-state combined FSTD and channel trellis releases two quaternary symbolsafter a delay equal to the path memory. The 10-state trellis is described in Table 4 where the (i,j)th location--i vertically and j horizontally indicated--provides the channel input and output symbol pairs associated with the transition from trellis state i to trellis state j. For example, the transition from trellis state 3 to trellis state 5 is associated with the channel input symbol pair +1 -1 and the channel output symbol pair 2 -2. In a second step, a block decoder without look-ahead operating on groups of 6 quaternary symbols inverts the encoder mapping in Table 2 and releases 8 information bits.

An 8B/6Q code that requires only 8 trellis states for Viterbi detection hasalso been designed. The construction of this code is based on the u-state transition diagram shown in FIG. 4. This FSTD is a subdiagram of the FSTD shown in FIG. 2, obtained by eliminating four edges arriving at the centerstate labeled by (-1, +3), (+1, -3), (+3, -3), (-3, +3). The RDS of sequences generated by this FSTD takes still 7 different values as before.However, the capacity has now decreased to 1.431 bit/symbol. Applying inputstate splitting to the center state in the third power of the FSTD in FIG. 4, one can design an 8B/6Q encoder with reduced average transmitted power,no quasi-catastrophic error propagation and a corresponding block decoder without look-ahead. The structure of the resulting 4-state encoder is shown in FIG. 5, the encoder mapping is specified in Table 5 and, finally,the list of code words in hexadecimal representation is given in Table 6. The average transmitted power now is P_(s) =3.29 which is slightly higher than in the previous case. Decoding is performed on an 8-state trellis obtained by combining the FSTD in FIG. 4 and the trellis of the dicode channel. This trellis is described in Table 7. Finally, the path memory of the Viterbi detector is limited to only 24 symbols.

Codes based on G₁₁

A higher rate code that requires 12 trellis states for Viterbi detection has also been designed. The construction of this code is based on a subdiagram of the 5-state irreducible component of the second power of G₁₁, N=11. Elimination of the two corner states at both ends of this 5-state component leads to a u-state transition diagram shown in FIG. 6, wherein u-3. The RDS of sequences generated by this FSTD takes 11 different values and the capacity is 1.613 bit/symbol. Applying input state splitting to the center state in the third power of the FSTD in FIG.6, one can design an 9B/6Q encoder with reduced average transmitted power, no quasi-catastrophic error propagation, and a corresponding block decoderwithout look-ahead. The structure of the resulting 4-state encoder is shownin FIG. 7 and the encoder mapping is specified in Table 8. Subtable 8.1 provides data byte to code word assignment when the encoder is in state 1 or state 4 (present state), whereas Subtable 8.2 shows the same for encoder states 2 and 3. Finally, the list of code words in hexadecimal representation is given in Tables 9A and 9B. The average transmitted powernow is P_(s) =4.02 which is slightly higher than in the previous two cases, but still 20% lower than the average transmitted power of the uncoded baseline system. Decoding is performed on a 12-state trellis obtained by combining the FSTD in FIG. 6 and the trellis of the dicode channel. This trellis is described in Table 10. Finally, the path memory of the Viterbi detector is again limited to only 24 symbols.

For example, the 6-symbol code word +1 -3 +1 +3 -1 -1 has the hexadecimal representation 8B5 and belongs to the list of code words F(18) (see Table 9). The code word 8B5 in F(18) can be generated when the encoder is in state s=2 or s=3 (see Subtable 8.2 and FIG. 7). In both cases, the next state is state s=2 or s=3 as it can be seen from Subtable 8.2 and FIG. 7.

Simulation Results

Simulation results for the proposed codes have verified the expected codinggains. FIG. 8 shows event error probabilities, P_(E) as a function of theenergy-per-symbol to noise-power-density ratio scaled by the rate loss, 2E_(s/RND) in dB. Curve 1 in FIG. 8 gives the theoretical performance ofan MLSE (Maximum Likelihood Sequence Estimation) receiver for uncoded quaternary transmission over the dicode channel (R=2 bit/symbol) estimatedby ##EQU1##

Curves 2-4 in FIG. 8 correspond to the performance of receivers for coded quaternary transmission at rates R=8/6 bit/symbol and R=9/6 bit/symbol over the dicode channel.

Curve 2 shows the case of the rate R=9/6 bit/symbol code where the Viterbi detector operates on the 12-state combined trellis described in Table 10. Compared to curve 1, a gain in noise margin of 2.9 dB is obtained. Curve 3corresponds to a rate R=8/6 bit/symbol code where the Viterbi detector operates on the 8-state combined trellis in Table 7. Compared to curve 1, a gain of approximately 3.4 dB is obtained. Finally, curve 4 shows the case of a rate R=8/6 bit/symbol code where the receiver utilizes a 10-state Viterbi detector with a trellis described in Table 4. A comparison of curve 4 with curve 1 indicates a gain of 3.8 dB in noise margin.

It is also of interest to calculate the spectrum of the particular codes proposed here. This will shed some light on spectral characteristics of encoded sequences such as width of spectral notch at f=0 and average transmitted power. FIG. 9 shows the power spectral densities of the various codes described in the previous section versus the normalized frequency, f/f_(s), where f stands for frequency and f_(s) represents the symbol frequency, i.e. the inverse of the duration of a quaternary symbol. The area under the curves in FIG. 9 corresponds to the average transmitted power P_(s). It can be seen that all three codes have reduced average transmitted power, i.e. their average transmitted power isconsiderably less than the average transmitted power in the case of uncodedquaternary transmission. Finally, Table 11 summarizes the parameters associated with the three codes described in the previous section and compares them against the baseline system of uncoded quaternary transmission.

Implementation

The finite-state machine encoders and block decoders disclosed here can be implemented either by random logic or by Read Only Memory blocks (ROMs).

Referring now to FIGS. 10 and 11, a ROM-based realization of the two 8B/6Q finite-state machine encoders described in Tables 2 and 5 is shown. It consists of an 8-bit input register 1, a 2-bit state register 2, a 1k×14 ROM 3, and a converter 4. Registers 1 and 2 are clocked at a rate of f_(s) /6, where f_(s) is the symbol rate.

At the end of an encoder cycle, a 12-bit code word is generated which is serialized and mapped into 6 quaternary symbols in converter 4. Decoding of the 8B/6Q codes is accomplished by a Viterbi detector 5 in cascade witha block decoder 6 shown in FIG. 11.

Channel output samples are quantized to q bits. A serial-to-parallel converter, consisting of two q-bit registers 7 and 8 and a 2q-bit register9, supplies a Viterbi detector 5 with two q-bit channel output samples. Registers 7 and 8 are clocked at a rate f_(s), whereas register 9 is clocked at a rate f_(s) /2. Viterbi detector 5 accepts the two q-bit samples and delivers four bits corresponding to delayed estimates of two quaternary symbols. The decision delay is equal to the path memory of the Viterbi detector. Three consecutive 4-bit outputs of Viterbi detector 5 stored in registers 10, 11, and 12, respectively, are passed on to a 12-bit register 13 whose content is interpreted as the 12-bit address of a4k×8 ROM 6. This ROM 6 constitutes the block decoder. Registers 10, 11, and 12 are clocked at a rate f_(s) /2, whereas register 13 is clocked at a rate f_(s) /6. The content of an 8-bit output register 14 represents the transmitted data byte. The output register 14 is clocked ata rate f_(s) /6.

Referring now to FIGS. 12 and 13, a ROM-based realization of the 9B/6Q finite-state machine encoder described in Table 8 is shown. It consists ofa 9-bit input register 15, a 2-bit state register 16, a 2k×14 ROM 17,and a converter 18.

At the end of an encoder cycle, a 12-bit code word is generated which is serialized and mapped into 6 quaternary symbols by converter 18. Decoding of the 9B/6Q code is accomplished by a 12-state Viterbi detector 19 in cascade with a block decoder 20, here a 4k×9 ROM, shown in FIG. 13.

Channel output samples are quantized to q bits. A serial to parallel converter, consisting of two q-bit registers 21 and 22 and a 2q-bit register 23, supplies Viterbi detector 19 with two q-bit channel output samples. Registers 21 and 22 are clocked at a rate f_(s), whereas register 23 is clocked at a rate f_(s) /2. Viterbi detector 19 accepts the two q-bit samples and delivers four bits corresponding to delayed estimates of two quaternary symbols. The decision delay is equal to the path memory of the Viterbi detector. Three consecutive 4-bit outputs of Viterbi detector 19 stored in registers 24, 25, and 26, respectively, are passed on to a 12-bit register 27 whose content is interpreted as the 12-bit address of a 4k×9 ROM 20. This ROM 20 constitutes the block decoder. Registers 24, 25, and 26 are clocked at a rate f_(s) /2, whereas register 27 is clocked at a rate f_(s) /6. The content of a 9-bit output register 28 represents the transmitted data byte. The output register 28 is clocked at a rate f_(s) /6.

While the present invention has been particularly shown and described with reference to preferred embodiments thereof, it will be understood by thoseskilled in the art that various changes in form and detail may be made therein without departing from the spirit, scope and teaching of the invention. Accordingly, the invention herein disclosed is to be consideredmerely as illustrative and limited in scope only as specified in the appended claims.

                  TABLE 1                                                          ______________________________________                                         N       C               N     C                                                ______________________________________                                         4       1.0000          10    1.7573                                           5       1.2924          11    1.7933                                           6       1.4499          12    1.8218                                           7       1.5664          13    1.8447                                           8       1.6508          14    1.8635                                           9       1.7111          15    1.8790                                           ______________________________________                                    

                  TABLE 2                                                          ______________________________________                                         Subtable 2.1                                                                                  1            4                                                  ______________________________________                                          0-42          A(43)/2      .sup.-- A(43)/3                                     43-66         B(24)/2      .sup.-- B(24)/3                                     67-131        C(65)/3      .sup.-- C(65)/2                                    132-153        D(22)/1      .sup.-- D(22)/4                                    154-255        E(102)/4     .sup.-- E(102)/1                                   ______________________________________                                         Subtable 2.2                                                                                  2            3                                                  ______________________________________                                          0-42          A(43)/4      .sup.-- A(43)/1                                     43-54         F(12)/2      F(12)/2                                             55-66         .sup.-- F(12)/3                                                                             .sup.-- F(12)/3                                     67-131        G(65)/1      G(65)/1                                            132-196        .sup.-- G(65)/4                                                                             .sup.-- G(65)/4                                    197-255        H(59)/3      .sup.-- H(59)/2                                    ______________________________________                                    

                                      TABLE 3                                      __________________________________________________________________________     Encoder tables: 8B6Q 10-state VD                                               __________________________________________________________________________     List A(43)                                                                            799                                                                               697                                                                               A57                                                                               6D6                                                                               A5A                                                                               A5D                                                                               A66 A69                                                                               6D9                                                                               676                                                A96                                                                               A99                                                                               69A                                                                               A8A                                                                               69D                                                                               679                                                                               A2A 66D                                                                               6A6                                                                               6A9                                                967                                                                               96A                                                                               96D                                                                               75A                                                                               976                                                                               979                                                                               D5A 667                                                                               D66                                                                               D69                                                997                                                                               766                                                                               99A                                                                               99D                                                                               D96                                                                               D99                                                                               769 9A6                                                                               9A9                                                                               9D6                                                9D9                                                                               66A                                                                               796                                                               List B(24)                                                                            A2D                                                                               A27                                                                               C6A                                                                               C9A                                                                               CA6                                                                               CA9                                                                               A93 A63                                                                               78A                                                                               A87                                                A8D                                                                               A36                                                                               A39                                                                               72A                                                                               6CA                                                                               757                                                                               75D D57                                                                               D5D                                                                               6A3                                                D8A                                                                               9A3                                                                               D2A                                                                               9CA                                                            List C(65)                                                                            E95                                                                               E98                                                                               E92                                                                               E86                                                                               E89                                                                               E1A                                                                               E26 E29                                                                               7A5                                                                               A6C                                                7A8                                                                               A75                                                                               A78                                                                               7A2                                                                               A72                                                                               A9C                                                                               AA5 AA8                                                                               AA2                                                                               7D5                                                6E5                                                                               AD5                                                                               AD8                                                                               AD2                                                                               AC6                                                                               AC9                                                                               6E8 6E2                                                                               B56                                                                               B59                                                B65                                                                               B68                                                                               B62                                                                               B4A                                                                               B95                                                                               B98                                                                               B92 B86                                                                               B89                                                                               B1A                                                B26                                                                               B29                                                                               6AC                                                                               D75                                                                               6B5                                                                               DA5                                                                               DA8 DA2                                                                               9AC                                                                               DD5                                                9B5                                                                               9B8                                                                               775                                                                               9B2                                                                               6B8                                                                               E56                                                                               E59 9E5                                                                               E65                                                                               E68                                                9E8                                                                               E62                                                                               6B2                                                                               9E2                                                                               E4A                                                         List D(22)                                                                            AA4                                                                               AA1                                                                               B55                                                                               B49                                                                               B94                                                                               B91                                                                               B85 B16                                                                               B19                                                                               B25                                                DA4                                                                               DA1                                                                               E55                                                                               E58                                                                               E52                                                                               E64                                                                               E61 E46                                                                               E49                                                                               E94                                                E91                                                                               E85                                                                  List E(102)                                                                           D76                                                                               D79                                                                               AD9                                                                               9B9                                                                               67A                                                                               D97                                                                               D9A D9D                                                                               797                                                                               ACA                                                6DA                                                                               DA6                                                                               DA9                                                                               9D7                                                                               79A                                                                               9DA                                                                               9DD 69E                                                                               A2B                                                                               79D                                                A2E                                                                               DD6                                                                               6DD                                                                               DD9                                                                               9E6                                                                               66B                                                                               9E9 A3A                                                                               67D                                                                               75B                                                B57                                                                               7A6                                                                               B5A                                                                               6B6                                                                               B5D                                                                               7A9                                                                               75E B66                                                                               677                                                                               B69                                                6E6                                                                               E57                                                                               E5A                                                                               96B                                                                               E5D                                                                               A5B                                                                               6B9 A5E                                                                               E66                                                                               96E                                                E69                                                                               767                                                                               6E9                                                                               A67                                                                               B96                                                                               76A                                                                               B99 A6A                                                                               977                                                                               E96                                                A6D                                                                               E99                                                                               97A                                                                               6A7                                                                               B8A                                                                               E8A                                                                               97D A76                                                                               A79                                                                               76D                                                66E                                                                               7D6                                                                               6AA                                                                               7D9                                                                               E2A                                                                               A97                                                                               A9A B2A                                                                               99B                                                                               A9D                                                99E                                                                               776                                                                               AA6                                                                               69B                                                                               AA9                                                                               9A7                                                                               779 9AA                                                                               AA3                                                                               D5B                                                D5E                                                                               9AD                                                                               6AD                                                                               D67                                                                               A8B                                                                               D6A                                                                               CAA 6D7                                                                               D6D                                                                               A8E                                                9B6                                                                               AD6                                                                  List F(12)                                                                            A53                                                                               A47                                                                               A4A                                                                               A4D                                                                               A17                                                                               A1A                                                                               A1D D1A                                                                               D4A                                                                               71A                                                74A                                                                               C5A                                                                  List G(65)                                                                            5B4                                                                               5B1                                                                               5E4                                                                               5E1                                                                               674                                                                               671                                                                               6A4 6A1                                                                               6D4                                                                               6D1                                                6C5                                                                               755                                                                               758                                                                               752                                                                               764                                                                               761                                                                               746 749                                                                               794                                                                               791                                                785                                                                               716                                                                               719                                                                               725                                                                               974                                                                               971                                                                               9A4 9A1                                                                               9D4                                                                               9D1                                                9C5                                                                               A55                                                                               A58                                                                               A52                                                                               A64                                                                               A61                                                                               A46 A49                                                                               A94                                                                               A91                                                A85                                                                               A88                                                                               A82                                                                               A16                                                                               A19                                                                               A25                                                                               A28 A22                                                                               D55                                                                               D58                                                D52                                                                               D64                                                                               D61                                                                               D46                                                                               D49                                                                               D94                                                                               D91 D85                                                                               D16                                                                               D19                                                D25                                                                               C56                                                                               C59                                                                               C65                                                                               C95                                                         List H(59)                                                                            A95                                                                               A98                                                                               A92                                                                               A86                                                                               A89                                                                               756                                                                               759 A26                                                                               A29                                                                               765                                                795                                                                               675                                                                               D56                                                                               D59                                                                               D65                                                                               D62                                                                               678 726                                                                               D95                                                                               729                                                D98                                                                               672                                                                               D92                                                                               D86                                                                               D89                                                                               D26                                                                               D29 C66                                                                               C69                                                                               6A5                                                96C                                                                               C96                                                                               C39                                                                               975                                                                               CA5                                                                               6A8                                                                               978 972                                                                               6A2                                                                               99C                                                9A5                                                                               9A8                                                                               9A2                                                                               6D5                                                                               9D5                                                                               9D8                                                                               6D8 9D2                                                                               9C6                                                                               9C3                                                6D2                                                                               6C6                                                                               6C3                                                                               A56                                                                               A59                                                                               A5C                                                                               A65 A68                                                                               A62                                            __________________________________________________________________________

                                      TABLE 4                                      __________________________________________________________________________     0        1     2     3     4     5     6     7     8     9                     __________________________________________________________________________     0  -1 +1 +1 -1 +3 -3 -1 +3 +1 +1 +3 -1       +1 +3 +3 +1                          -2 +2 0  -2 +2 -6 -2 +4 0  0  +2 -4       0  +2 +2 -2                       1  -1 +1 +1 -1 +3 -3 -1 +3 +1 +1 +3 -1       +1 +3 +3 +1                          0  +2 +2 -2 +4 -6 0  +4 +2 0  +4 -4       +2 +2 +4 -2                       2  -1 +1 +1 -1 +3 -3 -1 +3 +1 + 1                                                                               +3 -1       +1 +3 +3 +1                          +2 +2 +4 -2 +6 -6 +2 +4 +4 0  +6 -4       +4 +2 +6 -2                       3  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                                      -1                          -6 +4 -4 0  -2 -4 -6 +6 -4 +2 -2 -2 0  -6 -4 +4 -2 0      0                                                                       -4                       4  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                                      -1                          -4 +4 -2 0  0  -4 -4 +6 -2 +2 0  -2 +2 -6 -2 +4 0  0      +2                                                                      -4                       5  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                                      -1                          -2 +4 0  0  +2 -4 -2 +6 0  +2 +2 -2 +4 -6 0  +4 +2 0      +4                                                                      -4                       6  -3 +1 -1 -1 +1 -3 -3 +3 -1 +1 +1 -1 +3 -3 -1 +3 +1 +1     +3                                                                      -1                          0  +4 +2 0  +4 -4 0  +6 +2 +2 +4 -2 +6 -6 +2 +4 +4 0      +6                                                                      -4                       7        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                                      -1                                - 6                                                                               +2 -4 -2       -6 +4 -4 0  -2 -4 -6 +6 -4 +2     -2                                                                      -2                       8        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                                      -1                                -4 +2 -2 -2       -4 +4 -2 0  0  -4 -4 +6 -2 +2     0                                                                       -2                       9        -3 -1 -1 -3       -3 +1 -1 -1 +1 -3 -3 +3 -1 +1     +1                                                                      -1                                -2 +2 0  -2       -2 +4 0  0  +2 -4 -2 +6 0  +2     +2                                                                      -2                       __________________________________________________________________________

                  TABLE 5                                                          ______________________________________                                         Subtable 5.2                                                                                  1            4                                                  ______________________________________                                          0-31          A(32)/2      .sup.-- A(32)/3                                     32-47         B(16)/2      .sup.-- B(16)/3                                     48-94         C(47)/3      .sup.-- C(47)/2                                     95-140        D(46)/1      .sup.-- D(46)/4                                    141-255        E(115)/4     .sup.-- E(115)/1                                   ______________________________________                                         Subtable 5.1                                                                                  2            3                                                  ______________________________________                                          0-31          A(32)/4      .sup.-- A(32)/1                                     32-71         F(40)/4      .sup.-- F(40)/1                                     72-81         G(10)/2      G(10)/2                                             82-91         .sup.-- G(10)/3                                                                             .sup.-- G(10)/3                                     92-153        H(62)/1      H(62)/1                                            154-215        .sup.-- H(62)/4                                                                             .sup.-- H(62)/4                                    216-255        I(40)/3      .sup.-- I(40)/2                                    ______________________________________                                    

                                      TABLE 6                                      __________________________________________________________________________     Encoder tables: 8B6Q 8-state VD                                                __________________________________________________________________________     List A(32)                                                                            A96                                                                               66A                                                                               A69                                                                               A66 66D 96D                                                                               99D                                                                               D5A D5D                                                                               99A                                              D2A                                                                               D2D                                                                               9A9                                                                               6A9 6A6 D99                                                                               D96                                                                               D69 D66                                                                               9A6                                              6D9                                                                               6D6                                                                               96A                                                                               A5A A5D 69A                                                                               A2A                                                                               A2D 69D                                                                               9D9                                              9D6                                                                               A99                                                                  List B(16)                                                                            D8A                                                                               D8D                                                                               A8A                                                                               A8D 9CA 9CD                                                                               6CA                                                                               6CD C9A                                                                               C9D                                              C6A                                                                               C6D                                                                               CA9                                                                               CA6 CD9 CD6                                                    List C(47)                                                                            E4A                                                                               E4D                                                                               E1A                                                                               E1D E59 E56                                                                               AA5                                                                               AA2 E29                                                                               E26                                              AD5                                                                               E95                                                                               E92                                                                               AD2 E65 E62                                                                               9B5                                                                               A75 E35                                                                               A72                                              9B2                                                                               9E5                                                                               9E2                                                                               CB5 DA5 DA2                                                                               DD5                                                                               CE5 D75                                                                               D72                                              B4A                                                                               B4D                                                                               6B5                                                                               B1A B1D 6B2                                                                               B59                                                                               B56 6E5                                                                               B29                                              B26                                                                               6E2                                                                               B95                                                                               B92 B65 B62                                                                               B35                                                 List D(46)                                                                            AA4                                                                               AA1                                                                               AD4                                                                               AD1 A74 A71                                                                               DA4                                                                               DA1 D74                                                                               D71                                              B49                                                                               B46                                                                               B19                                                                               B16 B55 B52                                                                               B58                                                                               B25 B22                                                                               B28                                              B94                                                                               B91                                                                               B64                                                                               B61 E49 E46                                                                               E19                                                                               E16 E55                                                                               E52                                              E58                                                                               E25                                                                               E22                                                                               E28 E94 E91                                                                               E64                                                                               E61 9B4                                                                               9B1                                              9E4                                                                               9E1                                                                               6B4                                                                               6B1 6E4 6E1                                                    List E(115)                                                                           CA7                                                                               6DD                                                                               B69                                                                               B66 B63 AD9                                                                               AD6                                                                               6D7 B36                                                                               9AA                                              9AD                                                                               9A7                                                                               A79                                                                               A76 E4B E4E                                                                               A73                                                                               CDA 9B9                                                                               E1B                                              E1E                                                                               6B9                                                                               D5B                                                                               D5E 6B6 6B3                                                                               E5A                                                                               E5D E57                                                                               CB6                                              D2B                                                                               D2E                                                                               E2A                                                                               E27 9B6 66B                                                                               66E                                                                               9B3 E99                                                                               E96                                              D8B                                                                               D8E                                                                               CE6                                                                               E69 E66 E63                                                                               6E9                                                                               6E6 6E3                                                                               E36                                              D9A                                                                               D9D                                                                               D97                                                                               9CE 99B A5B                                                                               A5E                                                                               96B D6A                                                                               D6D                                              D67                                                                               96E                                                                               6CB                                                                               C9B C9E DA9                                                                               DA6                                                                               DA3 A2B                                                                               A2E                                              6CE                                                                               9E9                                                                               DD9                                                                               DD6 9E6 9E3                                                                               99E                                                                               A8B D79                                                                               D76                                              A8E                                                                               C6B                                                                               C6E                                                                               9DA 6AA B4B                                                                               B4E                                                                               6AD A9A                                                                               A9D                                              A97                                                                               6A7                                                                               B1B                                                                               B1E 9DD 9D7                                                                               A6A                                                                               B5A B5D                                                                               B57                                              A6D                                                                               A67                                                                               9CB                                                                               69B 69E B2A                                                                               B27                                                                               6DA AA9                                                                               AA6                                              AA3                                                                               B99                                                                               B96                                                                               CAA CAD                                                        List F(40)                                                                            757                                                                               75A                                                                               75D                                                                               72A 72D 727                                                                               799                                                                               796 793                                                                               769                                              766                                                                               763                                                                               739                                                                               736 979 976                                                                               973                                                                               679 676                                                                               697                                              667                                                                               A57                                                                               A27                                                                               D57 D27 967                                                                               997                                                                               A39 A36                                                                               D39                                              D36                                                                               9A3                                                                               A93                                                                               A63 9D3 D93                                                                               D63                                                                               6A3 6D3                                                                               673                                       List G(10)                                                                            A4A                                                                               A4D                                                                               A1A                                                                               A1D 8B5 8B2                                                                               8E5                                                                               8E2 2B5                                                                               2E5                                       List H(62)                                                                            8B4                                                                               8B1                                                                               8E4                                                                               8E1 9A4 9A1                                                                               9D4                                                                               9D1 974                                                                               971                                              6A4                                                                               6A1                                                                               6D4                                                                               6D1 674 671                                                                               A49                                                                               A46 A4C                                                                               A19                                              A16                                                                               A1C                                                                               A55                                                                               A52 A58 A25                                                                               A22                                                                               A28 A94                                                                               A91                                              A64                                                                               A61                                                                               A34                                                                               A31 D49 D46                                                                               D19                                                                               D16 D55                                                                               D52                                              D58                                                                               D25                                                                               D94                                                                               D91 D64 D61                                                                               749                                                                               746 719                                                                               716                                              755                                                                               752                                                                               758                                                                               725 794 791                                                                               764                                                                               761 5B4                                                                               5B1                                              5E4                                                                               5E1                                                                  List I(40)                                                                            A59                                                                               A56                                                                               A29                                                                               A26 A95 A92                                                                               A65                                                                               A62 A35                                                                               A32                                              D59                                                                               D56                                                                               9A5                                                                               9A2 D29 D26                                                                               9D5                                                                               D95 D92                                                                               9D2                                              D65                                                                               D62                                                                               975                                                                               D35 972 759                                                                               756                                                                               729 726                                                                               795                                              792                                                                               765                                                                               762                                                                               735 6A5 6A2                                                                               6D5                                                                               6D2 675                                                                               672                                       __________________________________________________________________________

                                      TABLE 7                                      __________________________________________________________________________     0       1     2     3     4     5     6     7                                  __________________________________________________________________________     0 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1                                      -2 +2 0  -2 +2 -6 0  0  +2 -4 0  +2 +2 -2                                    1 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1                                      0  +2 +2 -2 +4 -6 +2 0  +4 -4 +2 +2 +4 -2                                    2 -1 +1 +1 -1 +3 -3 +1 +1 +3 -1 +1 +3 +3 +1                                      +2 +2 +4 -2 +6 -6 +4 0  +6 -4 +4 + 2                                                                               +6 -2                                    3 -3 +1 -1 -1 +1 -3 -1 +1 +1 -1 -1 +3 +1 +1 +3 -1                                -4 +4 -2 0  0  -4 -2 +2 0  -2 -2 +4 0  0  +2 -4                              4 -3 +1 -1 -1 +1 -3 -1 +1 +1 -1 -1 +3 +1 +1 +3 -1                                -2 +4 0  0  +2 -4 0  +2 +2 -2 0  +4 +2 0  +4 -4                              5       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1                                      -6 +2 -4 -2 -6 +4 -4 0  -6 +6 -4 +2 -2 -2                              6       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1                                      -4 +2 -2 -2 -4 +4 -2 0  -4 +6 -2 +2 0  -2                              7       -3 -1 -1 -3 -3 +1 -1 -1 -3 +3 -1 +1 +1 -1                                      -2 +2 0  -2 -2 +4 0  0  -2 +6 0  +2 +2 -2                              __________________________________________________________________________

                  TABLE 8                                                          ______________________________________                                         Subtable 8.1                                                                                  1            4                                                  ______________________________________                                          0-138         A(139)/2     A(139)/3                                           139-229        B(91)/3      B(91)/2                                            230-301        C(72)/1      C(72)/4                                            302-511        D(210)/4     D(210)/1                                           ______________________________________                                         Subtable 8.2                                                                                  2            3                                                  ______________________________________                                          0-138         A(139)/4     .sup.-- A(139)/1                                   139-145        E(7)/4       .sup.-- E(7)/1                                     146-163        F(18)/2      F(18)/2                                            164-181        .sup.-- F(18)/3                                                                             .sup.-- F(18)/3                                    182-292        G(111)/1     G(111)/1                                           293-403        .sup.-- G(111)/4                                                                            .sup.-- G(111)/4                                   404-511        H(108)/3     H(108)/2                                           ______________________________________                                    

                                      TABLE 9A                                     __________________________________________________________________________     Encoder tables: 9B6Q 12-state VD                                               __________________________________________________________________________     List A(139)                                                                           63D                                                                               637                                                                               979                                                                               976                                                                               97C                                                                               75A                                                                               75D                                                                               757                                                                               6A9                                                                               39A                                                 39D                                                                               397                                                                               72A                                                                               72D                                                                               727                                                                               6A6                                                                               6AC                                                                               6A3                                                                               973                                                                               78A                                                 78D                                                                               787                                                                               36A                                                                               36D                                                                               367                                                                               799                                                                               796                                                                               79C                                                                               793                                                                               93A                                                 93D                                                                               6D9                                                                               769                                                                               766                                                                               76C                                                                               763                                                                               6D6                                                                               7C9                                                                               7C6                                                                               6DC                                                 6D3                                                                               739                                                                               736                                                                               679                                                                               676                                                                               3A9                                                                               3A6                                                                               67C                                                                               673                                                                               937                                                 3D9                                                                               3D6                                                                               99A                                                                               379                                                                               376                                                                               96A                                                                               A5A                                                                               A5D                                                                               A57                                                                               9CA                                                 9A9                                                                               C9A                                                                               C9D                                                                               A2A                                                                               A2D                                                                               A27                                                                               C97                                                                               9A6                                                                               69A                                                                               69D                                                 A8A                                                                               A8D                                                                               A87                                                                               697                                                                               C6A                                                                               C6D                                                                               A99                                                                               A96                                                                               A9C                                                                               A93                                                 C67                                                                               9AC                                                                               9A3                                                                               A69                                                                               A66                                                                               A6C                                                                               A63                                                                               9CD                                                                               9C7                                                                               CCA                                                 AC9                                                                               AC6                                                                               66A                                                                               A39                                                                               A36                                                                               A33                                                                               66D                                                                               667                                                                               96D                                                                               9D9                                                 9D6                                                                               9DC                                                                               CA9                                                                               D5A                                                                               D5D                                                                               D57                                                                               CA6                                                                               CAC                                                                               CA3                                                                               6CA                                                 D2A                                                                               D2D                                                                               D27                                                                               6CD                                                                               6C7                                                                               CD9                                                                               CD6                                                                               D8A                                                                               D8D                                                                               D87                                                 9D3                                                                               D99                                                                               D96                                                                               D9C                                                                               D93                                                                               967                                                                               99D                                                                               C79                                                                               D69                                                                               D66                                                 D6C                                                                               D63                                                                               C76                                                                               DC9                                                                               DC6                                                                               997                                                                               63A                                                                               D39                                                                               D36                                             List B(91)                                                                            DA5                                                                               DA2                                                                               DA8                                                                               DD5                                                                               DD2                                                                               DD8                                                                               CE5                                                                               CE2                                                                               D75                                                                               D72                                                 D78                                                                               9B5                                                                               9B2                                                                               7A5                                                                               7A2                                                                               7A8                                                                               7D5                                                                               7D2                                                                               7D8                                                                               9B8                                                 775                                                                               772                                                                               778                                                                               B4A                                                                               B4D                                                                               B47                                                                               B1A                                                                               B1D                                                                               B17                                                                               B59                                                 B56                                                                               B5C                                                                               B53                                                                               B29                                                                               B26                                                                               6B5                                                                               6B2                                                                               B89                                                                               B86                                                                               B95                                                 B92                                                                               B98                                                                               6B8                                                                               B65                                                                               B62                                                                               B68                                                                               9E5                                                                               3B5                                                                               B35                                                                               B32                                                 BC5                                                                               9E2                                                                               635                                                                               3E5                                                                               E4A                                                                               E4D                                                                               E47                                                                               6E2                                                                               6E8                                                                               9E8                                                 E1A                                                                               E1D                                                                               E17                                                                               E59                                                                               E56                                                                               E5C                                                                               E53                                                                               E29                                                                               E26                                                                               E89                                                 E86                                                                               E95                                                                               E92                                                                               E98                                                                               E65                                                                               E62                                                                               E68                                                                               E35                                                                               E32                                                                               EC5                                                 AA5                                                                               AA2                                                                               AA8                                                                               AD5                                                                               AD2                                                                               AD8                                                                               A75                                                                               A72                                                                               A78                                                                               CB5                                                 CB2                                                                     List C(72)                                                                            AA4                                                                               AA1                                                                               AD4                                                                               AD1                                                                               A74                                                                               A71                                                                               DA4                                                                               DA1                                                                               DD4                                                                               DD1                                                 D74                                                                               D71                                                                               7A4                                                                               7A1                                                                               7D4                                                                               7D1                                                                               774                                                                               771                                                                               B49                                                                               B46                                                 B4C                                                                               B19                                                                               B16                                                                               B1C                                                                               B55                                                                               B52                                                                               B58                                                                               B25                                                                               B22                                                                               B28                                                 B85                                                                               B82                                                                               B88                                                                               B94                                                                               B91                                                                               B64                                                                               B61                                                                               B34                                                                               B31                                                                               E49                                                 E46                                                                               E4C                                                                               E19                                                                               E16                                                                               E1C                                                                               E55                                                                               E52                                                                               E58                                                                               E25                                                                               E22                                                 E28                                                                               E85                                                                               E82                                                                               E88                                                                               E94                                                                               E91                                                                               E64                                                                               E61                                                                               E34                                                                               E31                                                 9B4                                                                               9B1                                                                               9E4                                                                               9E1                                                                               6B4                                                                               6B1                                                                               6E4                                                                               6E1                                                                               CB4                                                                               CB1                                                 CE4                                                                               CE1                                                                  __________________________________________________________________________

                                      TABLE 9B                                     __________________________________________________________________________     Encoder tables: 9B6Q 12-state VD                                               __________________________________________________________________________     List D(210)                                                                           6BC 9E3 B1B                                                                               B1E                                                                               CDA CDD CD7                                                                               9EC D2B D2E                                           9A7 B5A B5D                                                                               B57                                                                               37A 93E 99E                                                                               6E9 D8B B2A                                           B2D B27 B8E                                                                               6E6                                                                               6E3 3B9 3B6                                                                               B8A B8D B87                                           C7A B9A D9D                                                                               D97                                                                               B99 B96 B93                                                                               B9C C77 6EC                                           63B 3E9 3E6                                                                               B69                                                                               B66 B63 B6C                                                                               D6A D6D D67                                           B39 B36 63E                                                                               99B                                                                               97A CB9 CB6                                                                               BC3 BC6 DCA                                           DCD DC7 A5B                                                                               A5E                                                                               E4B E4E 97D                                                                               977 69B C9B                                           D3A D37 E1B                                                                               E1E                                                                               C9E CE9 CE6                                                                               A2B A2E DA9                                           DA6 E5A E5D                                                                               E57                                                                               DA3 DAC 69E                                                                               96B 6AA E2A                                           E2D E27 DD9                                                                               DD6                                                                               DD3 6AD 6A7                                                                               A8B E8A E8D                                           E87 A8E 9CB                                                                               D79                                                                               D76 D73 E99                                                                               E96 E93 E9C                                           9CE 39B 39E                                                                               C6B                                                                               E69 E66 E63                                                                               E6C C6E 96E                                           A9A 75B 75E                                                                               E39                                                                               E36 A9D A97                                                                               D9A 9B9 9B6                                           EC9 EC6 66B                                                                               36B                                                                               72B 72E 36E                                                                               6DA A6A A6D                                           A67 6DD 78B                                                                               78E                                                                               6D7 66E 9B3                                                                               ACA ACD 79A                                           79D 797 AC7                                                                               9BC                                                                               9DD 9D7 93B                                                                               76A 76D 767                                           A3A A3D A37                                                                               9AA                                                                               7CA 67A 67D                                                                               AA9 AA6 AA3                                           AAC 677 73A                                                                               737                                                                               9AD 6CB 6CE                                                                               CAA AD9 7A9                                           7A6 7A3 7AC                                                                               AD6                                                                               AD3 ADC 3AA                                                                               3AD 7D9 7D6                                           3A7 CAD CA7                                                                               A79                                                                               A76 779 776                                                                               773 A73 A7C                                           9E9 9E6 6B9                                                                               6B6                                                                               6B3 B4B B4E                                                                               3DA D5B D5E                                    List E(7)                                                                             A3C AC3 ACC                                                                               C3A                                                                               3A3 3CA 33A                                               List F(18)                                                                            8B5 8B2 8B8                                                                               8E5                                                                               8E2 8E8 A4A                                                                               A4D A47 A1A                                           A1D A17 2B5                                                                               2B2                                                                               2B8 2E5 2E2                                                                               2E8                                            List G(111)                                                                           5B4 5B1 5E4                                                                               5E1                                                                               2B4 2B1 2E4                                                                               2E1 8B4 8B1                                           8E4 8E1 9A4                                                                               9A1                                                                               9D4 9D1 974                                                                               971 6A4 6A1                                           6D4 6D1 674                                                                               671                                                                               CA4 CA1 CD4                                                                               CD1 C74 C71                                           3A4 3A1 3D4                                                                               3D1                                                                               374 371 A49                                                                               A46 A4C A43                                           A19 A16 A1C                                                                               A13                                                                               A55 A52 A58                                                                               A25 A22 A28                                           A85 A82 A88                                                                               A94                                                                               A91 A64 A61                                                                               A34 A31 AC4                                           AC1 D49 D46                                                                               D4C                                                                               D43 D19 D16                                                                               D1C D13 D55                                           D52 D58 D25                                                                               D22                                                                               D28 D85 D82                                                                               D88 D94 D91                                           D64 D61 D34                                                                               D31                                                                               DC4 DC1 749                                                                               746 74C 743                                           719 716 71C                                                                               713                                                                               755 752 758                                                                               725 722 728                                           785 782 788                                                                               794                                                                               791 764 761                                                                               734 731 7C4                                           7C1                                                                     List H(108)                                                                           6A5 6A2 6A8                                                                               6D5                                                                               6D2 6D8 675                                                                               672 678 CA5                                           CA2 CA8 CD5                                                                               CD2                                                                               CD8 C75 C72                                                                               C78 3A5 3A2                                           3A8 3D5 3D2                                                                               3D8                                                                               375 372 378                                                                               A59 A56 A5C                                           A53 A29 A26                                                                               A2C                                                                               A23 A89 A86                                                                               A8C A83 A95                                           A92 A98 A65                                                                               A62                                                                               A68 9A5 9A2                                                                               A35 A32 A38                                           9A8 AC5 AC2                                                                               AC8                                                                               9D5 9D2 9D8                                                                               975 972 978                                           D59 D56 D5C                                                                               D53                                                                               D29 D26 D2C                                                                               D23 D89 D86                                           D8C D83 D95                                                                               D92                                                                               D98 D65 D62                                                                               D68 D35 D32                                           D38 DC5 DC2                                                                               DC8                                                                               759 756 75C                                                                               753 729 726                                           72C 723 789                                                                               786                                                                               78C 783 795                                                                               792 798 765                                           762 768 735                                                                               732                                                                               738 7C5 7C2                                                                               7C8                                            __________________________________________________________________________

                                      TABLE 10                                     __________________________________________________________________________     0      1   2     3     4   5     6     7   8   9     10  11                    __________________________________________________________________________     0  -3 +3                                                                              -1 +1                                                                              +1 -1 +3 -3 -1 +3                                                                              +1 +1 +3 -1     +1 +3                                                                              +3 +1                              -6 +6                                                                              -4 +2                                                                              -2 -2 0  -6 -4 +4                                                                              -2 0  0  -4     -2 +2                                                                              0  -2                           1  -3 +3                                                                              -1 +1                                                                              +1 -1 +3 -3 -1 +3                                                                              +1 +1 +3 -1     +1 +3                                                                              +3 +1                              -4 +6                                                                              -2 +2                                                                              0  -2 +2 -6 -2 +4                                                                              0  0  +2 -4     0 +2                                                                               +2 -2                           2  -3 +3                                                                              -1 +1                                                                              +1 -1 +3 -3 -1 +3                                                                              +1 +1 +3 -1     +1 +3                                                                              +3 +1                              -2 +6                                                                              0 +2                                                                               +2 -2 +4 -6 0 +4                                                                               +2 0  +4 -4     +2 +2                                                                              +4 -2                           3  -3 +3                                                                              -1 +1                                                                              +1 -1 +3 -3 -1 +3                                                                              +1 +1 +3 -1     +1 +3                                                                              +3 +1                              0 +6                                                                               +2 +2                                                                              +4 -2 +6 -6 +2 +4                                                                              +4 0  +6 -4     +4 +2                                                                              +6 -2                           4      -3 +1                                                                              -1 -1 +1 -3 -3 +3                                                                              -1 +1 +1 -1 +3 -3                                                                              -1 +3                                                                              +1 +1 +3 -1                            -6 +4                                                                              -4 0  -2 -4 -6 +6                                                                              - 4                                                                               +2 -2 -2 0 -6                                                                               -4 +4                                                                              -2 0  0 -4                      5      -3 +1                                                                              -1 -1 +1 -3 -3 +3                                                                              -1 +1 +1 -1 +3 -3                                                                              -1 +3                                                                              +1 +1 +3 -1                            -4 +4                                                                              -2 0  0  -4 -4 +6                                                                              -2 +2 0  -2 +2 -6                                                                              -2 +4                                                                              0  0  +2 -4                     6      -3 +1                                                                              -1 -1 +1 -3 -3 +3                                                                              -1 +1 +1 -1 +3 -3                                                                              -1 +3                                                                              +1 +1 +3 -1                            -2 +4                                                                              0  0  +2 -4 -2 +6                                                                              0  +2 +2 -2 +4 -6                                                                              0 +4                                                                               +2 0  +4 -4                     7      -3 +1                                                                              -1 -1 +1 -3 -3 +3                                                                              -1 +1 +1 -1 +3 -3                                                                              -1 +3                                                                              +1 +1 +3 -1                            0 +4                                                                               +2 0  +4 -4 0 +6                                                                               +2 +2 +4 -2 +6 -6                                                                              +2 +4                                                                              +4 0  +6 -4                     8          -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                                                              -3 +3                                                                              -1 +1 +1                                                                                 +3 -3                            -6 +2 -4 -2     -6 +4 -4 0  -2 -4                                                                              -6 +6                                                                              -4 +2 -2                                                                                 02 -6                 9          -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                                                              -3 +3                                                                              -1 +1 +1                                                                                 +3 -3                            -4 +2 -2 -2     -4 +4 -2 0  0 -4                                                                               -4 +6                                                                              -2 +2 0 -2                                                                               +2 -6                 10         -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                                                              -3 +3                                                                              -1 +1 +1                                                                                 +3 -3                            -2 +2 0  -2     -2 +4 0  0  +2 -4                                                                              -2 +6                                                                              0  +2 +2                                                                                 +4 -6                 11         -3 -1 -1 -3     -3 +1 -1 -1 +1 -3                                                                              -3 +3                                                                              -1 +1 +1                                                                                 +3 -3                            0  +2 +2 -2     0  +4 +2 0  +4 -4                                                                              0 +6                                                                               +2 +2 +4                                                                                 +6 -6                 __________________________________________________________________________

                                      TABLE 11                                     __________________________________________________________________________     Rate (R)                                                                               Net increase in                                                                         Average Power                                                                           Maximum #Viterbi                                                                             Path Memory                                                                           #Encoder                                                                              Decoder Window           (bit/symbol)                                                                           noise margin (dB)                                                                       (P.)     Zero Runlength                                                                         States                                                                               (symbols)                                                                             States (symbols)                __________________________________________________________________________     2/1     0        500      ∞  4    ∞                                                                               1      1                        8/6     34       329      5        8    24     4      6                        8/6     38       293      5       10    24     4      6                        9/6     29       402      5       12    24     4      6                        __________________________________________________________________________ 

We claim:
 1. A method for encoding k-bit data bytes into m-symbol code words, the symbols being selected from a quaternary alphabet and the rate k/m (bits per symbol) being less than 2, comprising the steps of:forming a v-state machine according to a v-state transition diagram generating a first set of quaternary symbol sequences, said first set of quaternary symbol sequences being a part of a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of the sequences of said second set assuming values from a finite set of integers, such that each of the v states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word, the number of consecutive identical symbols in any sequence of said m-symbol code words being limited to a first given value, and the running digital sum in a sequence of such m-symbol code words being limited to one of a plurality of second given values, thereby enforcing a spectral null in the code spectrum; encoding in said v-state machine a k-bit data byte into an m-symbol code word; and generating a next state of said v-state machine depending on its current state and the last encoded k-bit data byte.
 2. A method for decoding m-symbol code words by estimating from output samples of a partial response channel the k-bit data bytes encoded in a v-state machine formed according to a v-state transition diagram and generating a first set of quaternary symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v-states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word, the number of consecutive identical symbols in any sequence of said m-symbol code words being limited to a first given value, and the running digital sum in a sequence of such m-symbol code words being limited to one of a plurality of second given values, thereby enforcing a spectral null in the code spectrum, said method comprising the steps of:generating the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and mapping said generated m-symbol words into k-bit data bytes.
 3. A method for digital transmission of data over a partial response channel using a quaternary alphabet, comprising:(A) encoding k-bit data bytes to be transmitted into m-symbol code words by:forming a v-state machine according to a v-state transition diagram generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of the sequences of said second set assuming values from a finite set of integers, such that each of the v states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word, the number of consecutive identical symbols in any sequence of said m-symbol words being limited to a first given value, and the running digital sum in a sequence of such m-symbol codes words being limited to one of a plurality of second given values, thereby enforcing a spectral null in the code spectrum; encoding in said v-state machine a k-bit data byte into an m-symbol code word; and generating a next state of said v-state machine depending on its current state and the last encoded byte; (B) transmitting said encoded data over said channel; and (C) decoding said transmitted data by:generating the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and mapping said m-symbol words into k-bit data bytes.
 4. A method for storing data in a multi-level memory using a quaternary alphabet, comprising:(A) encoding k-bit data bytes to be stored into m-symbol code words by:forming a v-state machine according to a v-state transition diagram generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of the sequences of said second set assuming values from a finite set of integers, such that each of the v states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word, the number of consecutive identical symbols in any sequence of said m-symbol code words being limited to a first given value, and the running digital sum in a sequence of such m-symbol code words being limited to one of a plurality of second given values, thereby enforcing a spectral null in the code spectrum; encoding in said v-state machine a k-bit data byte into an m-symbol code word; and generating a next state of said v-state machine depending on its current state and the last encoded byte; (B) storing said encoded data in said multi-level memory; and (C) upon reading said memory, decoding said stored data by:generating the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and mapping said m-symbol words into k-bit data bytes.
 5. An encoder for encoding k-bit data bytes into m-symbol quaternary code words wherein a v-state machine formed according to a v-state transition diagram generates a first set of quaternary symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v states is associated with 2K transitions, each transition corresponding to an m-symbol quaternary code word, said encoder comprising:input means for receiving and storing at least one k-bit data byte to be encoded; encoding means for encoding said k-bit data bytes into 2 m-bit bytes, said 2 m-bit bytes being associated with transitions corresponding to said m-symbol quaternary code words; and converter means for converting said 2 m-bit bytes into m-symbol quaternary code words.
 6. An encoder as in claim 5 wherein said input means comprises an 8-bit register and said encoding means comprises memory means and a 2-bit state register, said 2-bit state register coupled to said memory means and to said 8-bit register.
 7. An encoder as in claim 6 wherein said 8-bit input register and said 2-bit state register are clocked at a rate fs/6 where fs is the symbol rate for data being input to said input means.
 8. A decoder for decoding m-symbol quaternary code words by estimating from received output samples of a partial response channel the k-bit data bytes encoded in a v-state machine formed according to a v-state transition diagram which generates a first set of symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v states is associated with 2K transitions, each transition corresponding to an m-symbol quaternary code word, said decoder comprising:detector means for generating from said received channel output samples the most likely sequence of m-symbol quaternary words that can be derived from said u-state transition diagram and transforming said m-symbol quaternary words into 2 m-bit words; and decoder means coupled to an output of said detector means for mapping said 2 m-bit words into k-bit data bytes.
 9. A decoder as in claim 8 further comprising input means coupled to said detector means said for receiving and converting from serial to parallel said received channel output samples.
 10. A decoder as in claim 9 wherein said input means comprises a pair of q-bit registers in series, each said q-bit register coupled to a 2 q-bit register in parallel with the other q-bit register, said 2 q-bit register coupling two q-bit channel output samples to said detector means, said pair of q-bit registers being clocked at a rate fs and said 2 q-bit register being clocked at a rate fs/2, fs being the symbol rate of said received channel output samples.
 11. A decoder as in claim 8 wherein said decoder means comprises block decoder means coupled to an output register, the output of said output register representative of said decoded k-bit data bytes.
 12. A decoder as in claim 11 wherein said block decoder comprises address means coupled between said detector means and memory means, said address means comprising a plurality of first registers coupled in series, each of said first registers being coupled to a second register in parallel with the other first registers, the output of said second register constituting an address for said memory means.
 13. A decoder as in claim 12 wherein said memory means comprises a read-only memory.
 14. A decoder as in claim 8 wherein said detector means comprises a Viterbi detector.
 15. A digital transmission system for transmission of data over a partial response channel utilizing a quaternary alphabet, said data encoded in a v-state machine formed according to a v-state transition diagram which generates a first set of quaternary symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v states is associated with 2K transitions, each transition corresponding to an m-symbol quaternary code word, said digital transmission system comprising:input means for receiving and storing at least one k-bit data byte to be encoded; encoding means for encoding said k-bit data bytes into 2 m-bit bytes, said 2 m-bit bytes being associated with transitions corresponding to said m-symbol quaternary code words; converter means for converting said 2 m-bit bytes into m-symbol quaternary code words; a partial response channel for transmission of sequences of said m-symbol quaternary code words; detector means for generating from received channel output samples the most likely sequence of m-symbol quaternary words that can be derived from said u-state transition diagram and transforming said m-symbol quaternary words into 2 m-bit words; and decoder means coupled to an output of said detector means for mapping said 2 m-bit words into k-bit data bytes.
 16. A storage system for storing data in a multi-level memory utilizing a quaternary alphabet, said data encoded in a v-state machine formed according to a v-state transition diagram which generates a first set of quaternary symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v states is associated with 2K transitions, each transition corresponding to an m-symbol quaternary code word, said storage system comprising:input means for receiving and storing at least one k-bit data byte to be encoded; encoding means for encoding said k-bit data bytes into 2 m-bit bytes, said 2 m-bit bytes being associated with transitions corresponding to said m-symbol quaternary code words; converter means for converting said 2 m-bit bytes into m-symbol quaternary code words; a multi-level memory means for storing sequences of said m-symbol quaternary code words; detector means for generating from received memory means output samples the most likely sequence of m-symbol quaternary words that can be derived from said u-state transition diagram and transforming said m-symbol quaternary words into 2 m-bit words; and decoder means coupled to an output of said detector means for mapping said 2 m-bit words into k-bit data bytes.
 17. A method for decoding m-symbol code words by estimating from output samples of a partial response channel the k-bit data bytes encoded in a v-state machine formed according to a v-state transition diagram and generating a first set of quaternary symbol sequences that are included in a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of said sequences of the second set assuming values from a finite set of integers, such that each of the v-states is associated with 2^(k) transistions, each transition corresponding to an m-symbol code word, said method comprising the steps of:generating and detecting the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and block decoding and mapping said generated and detected m-symbol words into k-bit data bytes.
 18. A method for digital transmission of data over a partial response channel using a quaternary alphabet, comprising:(A) encoding k-bit data bytes to be transmitted into m-symbol code words by:forming a v-state machine according to a v-state transition diagram generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of the sequences of said second set assuming values from a finite set of integers, such that each of the v states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word; encoding in said v-state machine a k-bit data byte into an m-symbol code word; and generating a next state of said v-state machine depending on its current state and the last encoded byte; (B) transmitting said encoded data over said channel; and (C) decoding said transmitted data by:generating and detecting the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and block decoding and mapping said generated and detected m-symbol words into k-bit data bytes.
 19. A method for storing data in a multi-level memory using a quaternary alphabet, comprising:(A) encoding k-bit data bytes to be stored into m-symbol code words by:forming a v-state machine according to a v-state transition diagram generating a first set of quaternary symbol sequences that is part of a second set of quaternary symbol sequences derived from a u-state transition diagram, the running digital sum of the sequences of said second set assuming values from a finite set of integers, such that each of the v states is associated with 2^(k) transitions, each transition corresponding to an m-symbol code word; encoding in said v-state machine a k-bit data byte into an m-symbol code word; and generating a next state of said v-state machine depending on its current state and the last encoded byte; (B) storing said encoded data in said multi-level memory; and (C) upon reading said multi-level memory, decoding said stored data by:generating and detecting the most likely sequence of m-symbol words that can be derived from said u-state transition diagram; and block decoding and mapping said generated and detected m-symbol code words into k-bit data bytes. 